In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.
Mots clés : finite volume method, upwinding, shallow water, Harten regularization, source terms, entropy-correction
@article{M2AN_2003__37_5_755_0, author = {Rebollo, Tom\'as Chac\'on and Delgado, Antonio Dom{\'\i}nguez and Fern\'andez Nieto, Enrique D.}, title = {An entropy-correction free solver for non-homogeneous shallow water equations}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {755--772}, publisher = {EDP-Sciences}, volume = {37}, number = {5}, year = {2003}, doi = {10.1051/m2an:2003043}, mrnumber = {2020863}, zbl = {1033.76032}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2003043/} }
TY - JOUR AU - Rebollo, Tomás Chacón AU - Delgado, Antonio Domínguez AU - Fernández Nieto, Enrique D. TI - An entropy-correction free solver for non-homogeneous shallow water equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 755 EP - 772 VL - 37 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003043/ DO - 10.1051/m2an:2003043 LA - en ID - M2AN_2003__37_5_755_0 ER -
%0 Journal Article %A Rebollo, Tomás Chacón %A Delgado, Antonio Domínguez %A Fernández Nieto, Enrique D. %T An entropy-correction free solver for non-homogeneous shallow water equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 755-772 %V 37 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003043/ %R 10.1051/m2an:2003043 %G en %F M2AN_2003__37_5_755_0
Rebollo, Tomás Chacón; Delgado, Antonio Domínguez; Fernández Nieto, Enrique D. An entropy-correction free solver for non-homogeneous shallow water equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 5, pp. 755-772. doi : 10.1051/m2an:2003043. http://www.numdam.org/articles/10.1051/m2an:2003043/
[1] Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Comput. Methods Appl. Mech. Engrg. 155 (1998) 49-72. | Zbl
, , and ,[2] Upwind Methods for Hyperbolic Conservation Laws with Source Terms. Comput. & Fluids 23 (1994) 1049-1071. | Zbl
and ,[3] An introduction to finite volume methods for hyperbolic systems of conservation laws with source, Actas Ecole CEA - EDF - INRIA, Free surface geophysical flows, 7-10 Octobre, INRIA Rocquencourt, France (2002).
,[4] A non-parameterized entropy correction for Roe's approximate Riemann solver. Numer. Math. 73 (1996) 169-208. | Zbl
and ,[5] Simulación bidimensional de flujos hidrodinámicos transitorios en gemotrías irregulares. Ph.D. thesis Universidad de Zaragoza (2000).
,[6] A flux-splitting solver for shallow watter equations with source terms. Int. J. Num. Methods Fluids 42 (2003) 23-55. | Zbl
, and ,[7] A family of stable numerical solvers for Shallow Water equations with source terms. Comput. Methods Appl. Mech. Engrg. 192 (2003) 203-225. | Zbl
, and ,[8] Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. & Fluids 32 (2003) 479-513. | Zbl
, and ,[9] Hyperbolic systems of conservation laws. Math. Appl. (1991). | MR | Zbl
and ,[10] Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, Verlag (1996). | MR | Zbl
and ,[11] On upstream differencing and Godunov-type scheme for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35. | MR | Zbl
, and ,[12] A steady-state capturing method for hyperbolic systems with geometrical source terms. M2AN Math. Model. Numer. Anal. 35 (2001) 631-645. | Numdam | Zbl
,[13] Central-upwind schemes for the saint-venant system. M2AN Math. Model. Numer. Anal. 36 (2002) 397-425. | Numdam | Zbl
and ,[14] New High-Resolution Central Schemes for Nonlinear Conservations Laws and Convection-Diffusion Equations. J. Comput. Phys. 160 (2000) 214-282. | Zbl
and ,[15] Le Veque and H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86 (1990) 187-210. | Zbl
[16] Le Veque, Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm. J. Comp. Phys. 146 (1998) 346-365. | Zbl
[17] A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | Zbl
and ,[18] Upwind differencing schemes for hyperbolic conservation laws with source terms. Nonlinear Hyperbolic Problems, C. Carraso, P.A. Raviart and D. Serre, Eds., Springer-Verlag, Lecture Notes in Math. 1270 (1986) 41-51. | Zbl
,[19] E F. Toro., Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer (1997). | MR | Zbl
[20] Estudio de esquemas descentrados para su aplicacion a las leyes de conservación hiperbólicas con términos fuente. Ph.D. thesis, Universidad de Santiago de Compostela (1994).
,[21] Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry. J. Comp. Phys. 148 (1999) 497-526. | Zbl
,[22] High-order schemes and entropy condition for nonlinear hyperbolic systems of conservations laws. Math. Comp. 50 (1988) 53-73. | Zbl
,[23] The Surface Gradient Method for the Treatment of Source Terms in the Sallow-Water Equations. J. Comput. Phys. 168 (2001) 1-25. | Zbl
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